A generalization of a theorem of Jacobson

Author:
Susan Montgomery

Journal:
Proc. Amer. Math. Soc. **28** (1971), 366-370

MSC:
Primary 16.58

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276272-5

MathSciNet review:
0276272

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Abstract: A well-known theorem of Jacobson asserts that a ring in which for each in must be commutative. This paper gives a description of a ring with involution in which the above condition is imposed only on the symmetric elements. In particular, if is primitive, is either commutative or the matrices over a field, and, in general, any such is locally finite and satisfies a polynomial identity of degree 8.

**[1]**W. E. Baxter and W. S. Martindale III,*Rings with involution and polynomial identities*, Canad. J. Math.**20**(1968), 465–473. MR**0222116**, https://doi.org/10.4153/CJM-1968-043-6**[2]**I. N. Herstein,*Special simple rings with involution*, J. Algebra**6**(1967), 369–375. MR**0210747**, https://doi.org/10.1016/0021-8693(67)90089-0**[3]**I. N. Herstein,*Topics in ring theory*, The University of Chicago Press, Chicago, Ill.-London, 1969. MR**0271135****[4]**I. N. Herstein and Susan Montgomery,*A note on division rings with involutions*, Michigan Math. J.**18**(1971), 75–79. MR**0283017****[5]**Nathan Jacobson,*Structure of rings*, American Mathematical Society, Colloquium Publications, vol. 37, American Mathematical Society, 190 Hope Street, Prov., R. I., 1956. MR**0081264****[6]**Wallace S. Martindale III,*Jordan homomorphisms of the symmetric elements of a ring with involution*, J. Algebra**5**(1967), 232–249. MR**0210750**, https://doi.org/10.1016/0021-8693(67)90037-3**[7]**Wallace S. Martindale III,*Rings with involution and polynomial identities*, J. Algebra**11**(1969), 186–194. MR**0234990**, https://doi.org/10.1016/0021-8693(69)90053-2**[8]**J. Marshall Osborn,*Jordan algebras of capacity two*, Proc. Nat. Acad. Sci. U.S.A.**57**(1967), 582–588. MR**0215892****[9]**-, (to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276272-5

Keywords:
Rings with involution,
commutativity,
polynomial identity,
algebraic algebras

Article copyright:
© Copyright 1971
American Mathematical Society